Infinite products are an attractive part of real analysis which has fallen out of many syllabuses. I am concerned here only with infinite products in which the factors are between 0 and 1. The partial products are positive and decreasing, and so tend to a limit; the product is said to converge if the limit is non-zero, or diverge if it is zero. (Take logarithms to see why.)
For example, the infinite product ∏(1−1/n) (over all n≥2) diverges. Yesterday, in the course of a mistaken calculation (I was calculating the wrong thing), I noticed that ∏(1−1/n2) converges to 1/2.
Problem: What is ∏(1−1/nk) for k≥3?
I would be quite happy to be told that this is well known! As I said, I don’t actually need the answer …